An Exploration of the Physics Behind Rail Guns
Daniel Lenord
University of Alaska Fairbanks
Physics 212 Web Project, Spring 2003

    The speed of a rail gun slug is determined by several factors; the applied force, the amount of time that force is applied, and friction.  Friction will be ignored in this discussion, as it's effects can only be determined through testing. If this concerns you, assume a friction force equal to 25% of driving force.   The projectile, experiencing a net force as described in the above section, will accelerate in the direction of that force as in equation 1.

     (1) a=F/m

• a=Acceleration (Meters/second^2)

• F=Force on projectile (Newtons)

• m=Mass of projectile (Kilograms)

Unfortunately, as the projectile moves, the magnetic flux through the circuit is increasing and thus induces a back EMF (Electro Magnetic Field) manifested as a decrease in voltage across the rails.  The theoretical terminal velocity of the projectile is thus the point where the induced EMF has the same magnitude as the power source voltage, completely canceling it out.  Equation 2 shows the equation for the magnetic flux.

     (2) H=BA

• H=Magnetic Flux (Teslas x Meter^2)

• B=Magnetic field strength (Teslas) (Assuming uniform field)

• A=Area (Meter^2)

Equation 3 shows how the induced voltage V(i) is related to H and the velocity of the projectile.

     (3) V(i)=dH/dt=BdA/dt=BLdx/dt

• V(i)=Induced voltage

• dH/dt=Time rate of change in magnetic flux

• B=Magnetic field strength (Teslas)

• dA/dt=Time rate of change in area

• L=Width of rails (Meters)

• dx/dt=Time rate of change in position (velocity of projectile)

Since the projectile will continue to accelerate until the induced voltage is equal to the applied, Equation 4 shows the terminal velocity v(max) of the projectile.

     (4) v(max)=V/(BL)

• v(max)=Terminal velocity of projectile (Meters/second)

• V=Power source voltage (Volts)

• B=Magnetic field strength (Teslas)

• L=Width of rails (Meters)

These calculations give an idea of the theoretical maximum velocity of a rail gun projectile, but the actual muzzle velocity is dictated by the length of the rails.  The length of the rails governs how long the projectile experiences the applied force and thus how long it gets to accelerate.  Assuming a constant force and thus a constant acceleration, the muzzle velocity (assuming the projectile is initially at rest) can be found using Equation 5.

     (5) v(muz)=(2DF/m)^.5=(2DILB/m)^.5=I(2DLu/m)^.5

• v(muz)=Muzzle velocity (Meters/Second)

• D=Length of rails (Meters)

• F=Force applied (Newtons)

• m=Mass of projectile (Kilograms)

• I=Current through projectile (Amperes)

• L=Width between rails (Meters)

• B=Magnetic field strength (Teslas)

• u=1.26x10^-6 (The magnetic permeability of free space, Henries/Meter)

These calculations ignore friction and air drag, which can be formidable at the speeds and forces applied to the rail gun slug.  Top rail gun designs currently can launch a 2kg projectile with a muzzle velocity of close to 4km/s on roughly 6 meter rails. To reach this kind of velocity, the power source must provide roughly 6.5 million Amps.  Ouch.